Related papers: Kinematic generation of Darboux cyclides

Motivated by potential applications in architecture, we study Darboux cyclides. These algebraic surfaces of order a most 4 are a superset of Dupin cyclides and quadrics, and they carry up to six real families of circles. Revisiting the…

The induced metrics of Dupin cyclidic systems, that is, orthogonal coordinate systems with Dupin cyclides and spheres as coordinate surfaces, were provided by Darboux. Here we take a more geometric point of view and discuss how Dupin…

A surface that is the pointwise sum of circles in Euclidean space is either coplanar or contains no more than 2 circles through a general point. A surface that is the pointwise product of circles in the unit-quaternions contains either 2,…

Dupin cyclides are interesting algebraic surfaces used in geometric design and architecture to join canal surfaces smoothly and to construct model surfaces. Dupin cyclides are special cases of Darboux cyclides, which in turn are rather…

This article is devoted to the study of cyclides osculating general surfaces. We show that generically, at any point of a surface, one has a one-parameter family of cyclides tangent to a surface curve of order three and among them just one…

We study generators and relations of Cox rings of K3 surfaces of Picard number two. In particular we consider the Cox rings of classical examples of K3 surfaces, such as quartic surfaces containing a line and elliptic K3 surfaces.

We introduce a particular family of two-dimensional surfaces in $\mathbb R^4$ which generalize the classical Dini surfaces in $\mathbb R^3$.

The focus is on circular nets with one or two families of spherical parameter lines, which are treated in M\"obius geometry. These circular nets provide a discretisation of surfaces with one or two families of spherical curvature lines. The…

We prove that the cyclicity of a quadratic slow-fast integrable system of Darboux type with a double heteroclinic loop is finite and uniformly bounded.

We describe convex quadric surfaces in n dimensions and characterize them as convex surfaces with quadric sections by a continuous family of hyperplanes.

In this article we classify quadruple Galois canonical covers of smooth surfaces of minimal degree. The classification shows that they are either non-simple cyclic covers or bi-double covers. If they are bi-double then they are all fiber…

We describe a procedure for generating families of cyclic cubic fields with explicit fundamental units. This method generates all known families and gives new ones.

The task of recognizing an algebraic surface from a single apparent contour can be reduced to the recovering of a homogeneous equation in four variables from its discriminant. In this paper, we use the fact that Darboux cyclides have a…

We study the tropical lines contained in smooth tropical surfaces in R^3. On smooth tropical quadric surfaces we find two one-dimensional families of tropical lines, like in classical algebraic geometry. Unlike the classical case, however,…

In 1872 G. Darboux defined a family of curves on surfaces of R^3 which are preserved by the action of the Mobius group and share many properties with geodesics. Here we characterize these curves under the view point of Lorentz geometry and…

With the $[0,1,2]$-family of cyclic triangulations we introduce a rich class of vertex-transitive triangulations of surfaces. In particular, there are infinite series of cyclic $q$-equivelar triangulations of orientable and non-orientable…

We provide explicit equations of some smooth complex quartic surfaces with many lines, including all 10 quartics with more than 52 lines. We study the relation between linear automorphisms and some configurations of lines such as twin lines…

We construct motivic cohomology cycles in the group $H^3_{\mathcal M}(Z,{\mathbb Q}(2))$ where $Z$ is a K3 surface obtained as a double cover of a del Pezzo surface $X$ branched at a curve in $|-2K_X|$. The construction uses (-1) curves on…

We provide explicit generators for the torsion of the second cohomology of bielliptic surfaces, and we use this to study pullback map between Brauer group of a bielliptic surface and that of its canonical cover.

The purpose of this article is to give a geometric interpretation to the so-called "twelve surfaces of Darboux", or "Darboux wreath", which appear by applying repeatedly certain simple transformations to a given infinitesimal isometric…